76.10.20 problem 20

Internal problem ID [17471]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 20
Date solved : Monday, March 31, 2025 at 04:14:30 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-\frac {x \left (t \right )^{3}}{5}-\frac {y \left (t \right )}{5} \end{align*}

✗ Maple

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = x(t)-1/5*x(t)^3-1/5*y(t)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],t]==y[t],D[y[t],t]==x[t]-1/5*x[t]^3-1/5*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(x(t)**3/5 - x(t) + y(t)/5 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError :